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### Modelling elliptically polarised free electron lasers

View the table of contents for this issue, or go to the journal homepage for more 2016 New J. Phys. 18 062003

(http://iopscience.iop.org/1367-2630/18/6/062003)

FAST TRACK COMMUNICATION

## Modelling elliptically polarised free electron lasers

J R Henderson1,2,4

, L T Campbell1,2

, H P Freund3

and B W J McNeil1,5

1 _{SUPA, Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK}
2 _{ASTeC, STFC Daresbury Laboratory and Cockcroft Institute, Warrington, WA4 4AD, UK}
3 _{Colorado State University, Fort Collins, Colorado, 80523, USA}

4 _{Present address: Lancaster University, Engineering, LA1 4YR, UK.}
5 _{Author to whom any correspondence should be addressed.}

E-mail:j.r.henderson@strath.ac.uk,lawrence.campbell@strath.ac.ukandb.w.j.mcneil@strath.ac.uk

Keywords:free electron laser, accelerator, undulator radiation

### Abstract

### A model of a free electron laser

### (FEL)

### operating with an elliptically polarised undulator is presented.

### The equations describing the FEL interaction, including resonant harmonic radiation

### ﬁelds, are

### averaged over an undulator period and generate a generalised Bessel function scaling factor, similar to

### that of planar undulator FEL theory. Comparison between simulations of the averaged model with

### those of an unaveraged model show very good agreement in the linear regime. Two unexpected results

### were found. Firstly, an increased coupling to harmonics for elliptical rather than planar polarisarised

### undulators. Secondly, and thought to be unrelated to the undulator polarisation, a signiﬁcantly

### different evolution between the averaged and unaveraged simulations of the harmonic radiation

### evolution approaching FEL saturation.

### 1. Introduction

The free-electron laser(FEL)is a proven source of high-power tunable radiation over a wide spectral range into the hard x-ray[1], where its output is transforming our ability to investigate matter and how it functions, in particular in biology[2]. In addition to the atomic spatiotemporal resolution offered by the short wavelengths and pulses, the FEL can also generate radiation output from planar through to full circular polarisation using undulators of variable ellipticity such as the APPLE-III undulator design, proposed for SwissFEL[3], and the Delta undulator design[4], installed at LCLS[5]. This variably polarised output offers another important degree of freedom with which to investigate the behaviour of matter and is of signiﬁcant interest across a wide range of science[6–9]. FEL user facilities, such as the FERMI user facility in Italy, are now recognising and addressing this need for elliptically polarised output[10,11].

In a planar undulator, the electrons have a fast axial‘jitter’motion at twice the undulator period as they propagate along the undulator axis. In addition to the coupling of the electrons to the fundamental radiation wavelength, the jitter motion allows coupling to odd harmonics of the fundamental, which can also experience gain. A commonly used model used for simulating the FEL interaction is the‘averaged’model which, as the name suggests, averages the governing Maxwell and Lorentz equations describing the electron/radiation coupling over an undulator period[12]. The averaging of the jitter motion introduces coupling terms described by a difference of Bessel functions which depend upon both the undulator strength and the harmonic[12,13]. For an helical undulator, there is no electron jitter and the difference of Bessel functions coupling terms become a constant for the fundamental and zero for all harmonics, i.e in an helical undulator there is no gain coupling to harmonics.

It is perhaps surprising that the equivalent coupling terms for an elliptically polarised undulator have not been derived previously. In this paper, the coupling terms due to electron jitter motion are calculated in a general way for all undulator ellipticities from a planar through to an helical conﬁguration, corresponding to those now

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23 March 2016

REVISED

23 May 2016

ACCEPTED FOR PUBLICATION

6 June 2016

PUBLISHED

20 June 2016

Original content from this work may be used under the terms of theCreative Commons Attribution 3.0 licence.

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available from variably polarised undulators, so enabling more accurate modelling of this important type of FEL output.

The resultant derived coupling terms, which are a more general form of the difference of Bessel functions factors of the planar undulator case, are used to predict the scaling of the FEL interaction for a range of undulator ellipticities. An averaged FEL simulation code then uses the general Bessel function factors to give solutions of elliptically polarised FEL output into the nonlinear, high-gain regime and tested against the scaling. A further test is also made by comparing the results of the averaged FEL simulations with an unaveraged simulation code, Pufﬁn[14]. New, perhaps unexpected, results are presented and discussed.

### 2. The elliptical undulator model

In this section the equations describing the electron beam and radiation evolution in an elliptically polarised undulator are derived in the 1D limit. The equations are averaged over an undulator period removing any sub-wavelength information or effects such as coherent spontaneous emission.

The undulator magneticﬁeld with variable ellipticity is simply deﬁned as:

= -*B* (*k z*) ˆ +*u B* (*k z*) ˆ ( )

**B***u* 0sin *u* **x** *e* 0cos *u* **y**, 1

whereuedescribes the undulator ellipticity,B0the peak undulator magneticﬁeld, and*ku*=2*p lu*where*lu*is

the undulator period. The undulator ellipticity parameter varies in the range0*ue*1, from a planar(ue=0)

through to an helical undulator(u_{e}=1)to give an rms elliptical undulator parameter of:

= +

¯ ( )

*a* 1 *u* *a*

2 , 2

*u* *e* *u*

2

where the peak undulator parameter is deﬁned as*au*=*eB mck*0 *u*. The resonant fundamental FEL wavelength is

then:

*l* *l*
*g*

= ( +*a*¯ ) ( )

2 1 , 3

*r* *u*

*r*
*u*
2

2

where the resonant electron energy in units of electron rest mass*g _{r}*= á ñ

*g*, the mean of the electron beam.

2.1. The electron equations

In the averaged FEL model the electron orbits areﬁrst calculated in the absence of any radiationﬁeld from the Lorentz force equation:

**b**

**b**

*g*

= - ´* B* ( )

*t*

*e*
*m*
d

d , 4

*j*

*j*

*j* *u*

where* bj*=

**v**

*j*

*c*and

*gj*are thejth electron’s velocity scaled with respect to the speed of lightc, and the

corresponding Lorentz factor. Substituting for the undulatorﬁeld(1), and integrating the Lorentz equation(4), the scaled electron velocity components are obtained:

*b*
*g*
=
+
¯
( ) ( )
⎛
⎝
⎜ ⎞
⎠
⎟
*u*
*u*
*a*
*k z*
2

1 sin , 5

*xj* *e*
*e*
*u*
*j*
*u*
2
2
1 2
*b*
*g*
=
-+
¯
( ) ( )
⎛
⎝
⎜ ⎞
⎠
⎟
*u*
*a*
*k z*
2

1 cos , 6

*yj*
*e*
*u*
*j*
*u*
2
1 2
*b* *b*
*g*
= -
-+
¯ ¯ _{(} _{)} _{( )}
⎡
⎣
⎢
⎢
⎛
⎝
⎜ ⎞
⎠
⎟ ⎤
⎦
⎥
⎥
*u*
*u*
*a*
*k z*
1

1 cos 2 , 7

*zj* *z* *e*

*e*
*u*
*j*
*u*
2 2
2
2
2
1 2

where*v*¯*z* =*cb*¯*z*is the average longitudinal electron velocity. The constantsmandetake their usual meanings of

rest mass and charge magnitude of the electron. Introducing the non-unit vector basis**f**= 1_{2}(*ue***x**ˆ +i**y**ˆ), so
that**f f**· = -(1-*u _{e}*2) 2and

**f f**· *=(1+

*u*2) 2, the perpendicular components may be written:

_{e}**b***g*
=
+ -
-^ ¯ ( ( ) ) ( )
*u*
*a*
*k z*
**f**
i
1

exp i c.c. . 8

*j*
*e*
*u*
*j*
*u*
2
2

Integrating equation(7), the longitudinal electron trajectory in the presence of the undulatorﬁeld only is:

*b*

*g* *b* *b*

= - -+ ( ) ¯ ¯ ¯ ( ¯ ) ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

*z t* *c* *t* *a*

*k*

*u*

*u* *k c* *t*

4

1

1 sin 2 . 9

*j* *z* *u*

*j u z*

*e*
*e*
*u* *z*
2
2 2
2
2

The oscillatory term in(9)describes the‘ﬁgure-of-eight’longitudinal jitter motion of the electron in a non-helical undulator associated with coupling to harmonics of the radiationﬁeld[12].

A co-propagating radiationﬁeld is similarly deﬁned using the same non-unit vector basis**f**as the sum over
harmonics of the fundamental resonantﬁeld, i.e.**E**= å_{n}**E***n*, where:

= -*w*

-(* _{z t}*) (

_{}(

*) ( ) ) ( )*

_{z t}**E** , i **f**

2 , e c.c. . 10

*n* *n* i*n k zr* *rt*

The scaled energy evolution of thejth electron in the transverse plane-wave radiationﬁeld of(10)may then be written as:

### å

**b***g*= - ^ · ( )

*t*

*e*

*mc*

**E**d

d , 11

*j*

*n*

*j* *n*

using equations for the electron motion(8)and(9), the electricﬁeld(10)and the identity:

### å

=*f*

*f*=-¥ ¥ ( ) ( )

( ) _{J x}

e*x* e , 12

*n*

*n* *n*

i sin i

the equation for the electron energy(11)simpliﬁes to:

### å

### å

### å

*g*

*g*= -+ + + - +

*q*

*b*

*b*=-¥ ¥ - - + =-¥ ¥ - + + ¯ ( ) ( ) ( ) ( ) ( ) ( ) ¯ ( ) ¯ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜⎜ ⎛

_{⎝}⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟⎟

*t*

*e*

*mc*

*u*

*a*

*u* *J* *b*

*u* *J* *b*

d

d 4

2

1 e 1 e

1 e c.c. , 13

*j*

*e*
*u*

*j* *n*

*n* *n* *e*

*m*

*m* *n* *m k* *ct*

*e*
*m*

*m* *n* *m k* *ct*

2 1 2

i 2 i 1 2

2 i 1 2

*j* *u z*

*u z*

where*qj*=(*kr*+*ku*) ¯*bzct*-*wrt*is the ponderomotive phase. Resonant, non-oscillatory terms, which do not
average to zero over an undulator period occur only for*n* +1 2*m*=0, so that on averaging over an
undulator period equation(13)simpliﬁes further to:

### å

*g*

*g*= -+ +

*q*¯ ( ) ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

*t*

*e*

*mc*

*u*

*a*

*JJ*d d 4 2

1 e c.c. , 14

*j*

*e*
*u*

*j* *n*

*n* *n* *n*

2
1 2
i *j*
where:
*x* *x*
= - - + -
-- +

## (

## )

( ) ( ) ( ) ( ) ( ) ( )*JJn* 1 1 *u Je*2 1 *u Je*2 , 15

*n*
*n* *n*
1
2 1
2
1
2
*x*=
+
-+
¯
( ¯ ) ( )
*na*
*a*
*u*
*u*
2 1
1

1 . 16

*u*
*u*
*e*
*e*
2
2
2
2

2.2. The wave equation

The 1D wave equation is used to model the plane wave radiationﬁeld evolution and is given by:

*m*
*s*
¶
¶
-¶
¶ =
¶
¶
^
( )
⎛
⎝
⎜ ⎞
⎠
⎟

*z* *c* *t* **E** *t*

**J**
1
, 17
2
2 2
2
2
0

whereσis the transverse area of the co-propagating planar radiationﬁeld and electron beam with transverse
current density of**J**^= - å*ec* *j*= * b*^

*d*(

**r**-

**r**( ))

*t*

*N*

*j*

1 . The transverse components of the electricﬁeld and transverse

current density are deﬁned by*E*^= 2 **E f**· *and*J*^= 2 **J f**· *respectively. In the 1D limit, the wave

equation(17)simpliﬁes to:

*m*
*s*
¶
¶
-¶
¶ =
¶
¶
^ ^ ( )
⎛
⎝
⎜ ⎞_{⎠}⎟

*z* *c* *t* *E*

*J*
*t*
1
. 18
2
2 2
2
2
0

### å

### å

### å

*m*

*s*

*g*

*d*¶ ¶ + ¶ ¶ = + + + -

*-q*

*b*

*b*= -=-¥ ¥ - + =-¥ ¥ + + ¯ ( ) ( ) ( ) ( ) ( ) ( ( )) ( ) ( ) ¯ ( ) ¯ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

*z*

*c t*

*e* *c a*

*u* *u* *J* *b*

*u* *J* *b* *z* *z t*

1

2 1

e

1 e

1 e , 19

*n* *u*
*e* *j*
*N* *n*
*j*
*e*
*m*

*m* *n* *m k* *ct*

*e*
*m*

*m* *n* *m k* *ct* *j*

0 2 2 3 2

1 i

2 i 1 2

2 i 1 2

*j*

*u z*

*u z*

where*q*=(*kr*+*k zu*) -*wrt*is the ponderomotive phase of the fundamental wavelength. Resonant terms are
only seen to occur for*n* +1 2*m*=0and, asmis integer, the harmonic numbersnare therefore odd.
Applying this resonant condition yields:

### å

*m*

*s* *g* *d*

¶
¶ +
¶
¶ = +
*-q*
=
-¯
( ) ( ( )) ( )
⎜ ⎟
⎛
⎝
⎞
⎠
*z* *c t*

*e* *c a*

*u* *JJ* *z* *z t*

1
2 1
e
. 20
*n* *u*
*e* *j*
*N* *n*
*j*
*n* *j*
0 2
2 3 2

1 i

2.3. The scaled FEL model

The scaling of[15,16]is now applied using the FEL parameter*r*=*g*-* _{r}*1( ¯

*auwp*4

*cku*)2 3, where

*wp*is the peak

non-relativistic plasma frequency of the electron beam. The wave equation forﬁeld(20)is also averaged over a
radiation wavelength by assuming theﬁeld envelope does not change in this interval. The independent variables
are the scaled distance through the FEL*z*¯=*z lg*, and scaled position in the electron beam rest-frame

*b* *b* *rq*

= - =

¯ ( ¯ ) ¯

*z*1 *z* *c* *zt* *z cl* 2 *j*, where*lg* =*lu* 4*pr*and*lc*=*lr* 4*pr*are respectively the gain length and

cooperation length of the FEL interaction at the fundamental(n=1)in an helical undulator(u_{e}=1)[16].
Clearly, and as shown from the scaling below, these lengths are different for interactions at harmonics and in an
elliptical undulator.

Introducing the scaled harmonic radiation envelopes:

*rg*

= + ¯

( ) ( )

*A* *u* *a e*

*mc k*
1

2 4 , 21

*n* *e* *u* *n*

*u* *r*

2

2 2

the scaled electron energy*p _{j}* =(

*g*-

_{j}*g rg*)

_{r}*and using the deﬁnition of the ponderomotive phaseθ, the scaled equations for the 1D FEL interaction in an elliptically polarised undulator including harmonic radiationﬁelds are given by:*

_{r}*q*
=

¯ ( )

*z* *p*

d

d , 22

*j*
*j*

### å

*a*

= - *q* +

¯ ( ) ( )

*p*

*z* *A*

d

d e c.c. , 23

*j*
*n*
*n* *n*
,odd
i*j*
*a c*
¶
¶ +
¶

¶ = á -*q*ñ

¯ ¯ ( ¯ ) ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

*z* *z*1 *An* *n* *z* e , 24

1 i*j*

where*an*are ellipticity dependent coupling parameters given by:

*a* =

+ ( )

*JJ*
*u*

1 , 25

*n* *n*

*e*2

and*c*( ¯ )*z*1 =*I z*( ¯ )1 *Ipk*is the beam current scaled with respect to its peak value[13]. There is one wave equation

of type(24)for each harmonic considered. Notice from(21), that the harmonicﬁeld envelopes*n*are scaled so

that the∣*An*∣2are proportional to the power of the elliptically polarised harmonic radiationﬁelds over the full
range ofue, from planar to helical polarisation.

### 3. Modelling the elliptical undulator FEL

The equations for the elliptical model(22)–(24)are now solved for a range of ellipticity parametersue. The

solutions are determined by the ellipticity and harmonic dependent coupling parameters*an*which are speciﬁed

and used in scaling to predict the gain length and saturation powers of the elliptical FEL interaction.

Numerical solutions of the averaged elliptical FEL model of above are also compared with the unaveraged model of‘Pufﬁn’[14]. As the equations of this model are unaveraged, no factors such as(25)appear in the model and Pufﬁn can simulate the FEL interaction for an undulator of any ellipticity and over a broad radiation bandwidth that includes harmonic content.

3.1. Scaling

Figure1plots the elliptical coupling parameters*an*as a function of the ellipticity parameteruefor the resonant

odd harmonics*n*= ¼1 7and for a range of rms undulator parameters*a*¯*u*. The coupling parameters agree with

4

previous results in the helical and planar limits. It is worth noting that for the harmonicﬁelds*n*>1, and for
larger undulator parameters*a*¯*u*, that the coupling is stronger for elliptically polarised undulators rather than the

planar case ofue=0. This result is perhaps somewhat unexpected.

If the equations for the elliptical model(22)–(24)are written in the absence of any harmonic interactions, i.e.
forn=1 only, then the elliptical coupling parameter*a*1could be incorporated into the scaling to give a system of

universally scaled equations with no free parameters[15]. In this case the FEL scaling parameter would now
depend upon the elliptical coupling parameter for the fundamental as*r*µ*a*12 3, so that the gain length of the

interaction, and so also the saturation lengthzsat, would scale as*lg*,*z*satµ*a*-12 3. The scaled saturation power

would scale as∣ ∣*A*sat2 µ*a*12 3.

In the simulations which follow, an electron pulse of charge 70 pC is assumed with a uniform current,
*c*( ¯ )*z*1 =1, over scaled pulse length of*l*¯*e*=*l le* *c*=129. A mean beam energy*gr*=1500with zero energy spread

and an FEL parameter of*r*= ´2 10-3_{is used. Unless otherwise stated, the undulator has}_{ﬁ}_{xed rms undulator}

parameter of*a*¯*u*=1.0independent of the undulator ellipticity, to give aﬁxed resonant radiation wavelength of
*lr*=16nm. A seed laser of scaled amplitude of*A*0=10-4was used to initiate the FEL interaction. This

eliminates shot-to-shot variation of the radiation pulse saturation energy and saturation length which occurs when the interaction starts from noise, simplifying comparison with analysis and the results obtained from the solutions of the different numerical codes. The total scaled energy of an harmonic of the radiation pulse is deﬁned by:

Figure 1.The elliptical coupling parameters*an*plotted as a function of the ellipticity parameteruefor theﬁrst four odd harmonics

=

*n* 1, 3, 5, 7. Four different rms undulator parameters are shown in each graph:*a*¯*u*=0.5, 1.0, 2.5, 5.0. The*an*agree with previous

analysis in the helical and planar limits,ue=1 andue=0 respectively. Note that for larger undulator parameters*a*¯*u*, the coupling

parameters*an*for harmonics maximise for an elliptical undulator conﬁguration,*ue*>0. For example, for the third harmonic with

= ¯

### ò

=-¥ +¥

( ¯) ∣ ( ¯ ¯ )∣ ¯ ( )

*E zn* *A z zn* , 1 2d .*z*1 26

with the total given by the sum over the odd harmonics*E*= å* _{n odd}*,

*En*. As the electron pulse is many cooperation

lengths long(*l*¯*e*=129)and the interaction is seeded, the interaction will approximate a steady-state interaction
where pulse effects are small. In this case, the scaled pulse energy at saturation, either for a particular harmonic
componentnor for the total, will be*E*sat»*l A*¯ ∣ ∣*e* sat2 . For an helical undulator in the steady-state, the scaled

saturation power of the fundamental(n=1)is∣ ∣*A*sat2 »1.37. For the case considered here this gives a scaled
pulse energy at saturation of*E*sat»177.

In order to test the above scaling for the scaled saturation energy and saturation length, the equations(22)–

(24)were solved numerically for the above parameters in the absence of any harmonic interaction for a range of undulator ellipticities. Figure2demonstrates that the numerical solutions are in very good agreement with the predicted scaling.

3.2. Comparison between averaged and unaveraged models

Numerical solutions to the averaged elliptical model of equations(22)–(24)are now compared with the those generated by the unaveraged code Pufﬁn[14], which is able to model an FEL interaction in an elliptically polarised undulator across a broad bandwidth radiationﬁeld that includes harmonic content. The unaveraged electron motion of the Pufﬁn model includes any‘jitter’motion of equation(9)due to an elliptically polarised undulator.

As Pufﬁn is an unaveraged FEL simulator, the effects of self ampliﬁed coherent spontaneous emission can be signiﬁcant when modelling a‘ﬂat-top’electron bunch which has discontinuities in the electron beam current. As these effects cannot be modelled in an averaged model, the electron bunch used in the Pufﬁn simulations here is modiﬁed to have smooth ramp down in current over several radiation wavelengths at the electron bunch edges. This smooth ramping of the current signiﬁcantly reduces the generation of any coherent spontaneous emission, enabling a better comparison between the two models.

In what follows, only the fundamental and third harmonics(n=1, 3)are modelled using the above parameters. In the averaged model, the harmonic radiation content is obtained directly from the individual harmonic components,An. In the unaveraged model, however, access to the content of each harmonic is

obtained by fourierﬁltering the broadband radiationﬁeld about a narrow bandwidth of the particular harmonic of interest(in this case forn=3.)

Figure3plots the scaled pulse energy of the fundamentalE1, from the averaged and Pufﬁn simulations as a

function of scaled propagation distance through the interaction*z, for three different undulator ellipticities,*¯
=

*ue* 0, 0.5, 1.0. Excellent agreement between the simulations is seen for allue, well into the saturated,

nonlinear regime.

Figure 2.Comparison between numerical solutions of the averaged model of equations(22)–(24)in the absence of any harmonic
interactions(red crosses)and the predicted scaling with respect to the elliptical coupling parameter of the fundamental*a*1(blue line)

for the full range of the ellipticity from planar(ue=0)to helical(ue=1). The top plot shows the saturated pulse energyEsatand the

lower the scaled saturation length*z*¯sat.

6

The scaled radiation pulse energiesEnof the fundamental and third harmonic for both averaged and

unaveraged simulations for the planar undulator(u_{e}=0)are shown inﬁgure4. As previously seen inﬁgure3, the
fundamental pulse energiesE1of the averaged and unaveraged simulations are in excellent agreement. The third

harmonic shows reasonable agreement in the decoupled linear regime until*z*¯»11. At this point in the averaged
model, the electron bunching at the fundamental also begins to drive the third harmonicﬁeld with a growth rate
∼3 times that of the fundamental[17]. While there is evidence of similar enhanced harmonic growth in the
unaveraged simulation, the effect is seen to be signiﬁcantly less pronounced. As the interaction proceeds into the
nonlinear, saturation regime for*z*¯>13, both simulations are seen to resume a similar evolution.

It was noted fromﬁgure1that for larger undulator parameters*a*¯*u*, the coupling parameters*an*for

harmonics maximise for an elliptical undulator conﬁguration,*ue* >0. This increased coupling can be expected

to decrease the gain length and increase the saturation pulse energies of harmonics for these elliptical
polarisations. In particular, the gain length for the third harmonic in an undulator with parameter*a*¯*u*=5.0,

Figure 3.Simulations using the averaged and unaveraged models show excellent agreement for the evolution of the scaled radiation
pulse energy of the fundamentalE1, as a function of scaled distance through the undulator for planar_{(}blue,ue=0.0), elliptical(red,

ue=0.5)and helical(black,ue=1.0)undulator polarisation.

Figure 4.Comparison of the scaled pulse radiation energies*E*1,3for averaged and unaveraged simulations in a planar undulator

should be minimised for an elliptical undulator with*ue* »0.34. From the above scaling(and writing*l ug*( )*e* , etc)

the ratio of the two gain lengths*lg*(0.34) *lg*( )0 =0.934.

Both the averaged and unaveraged numerical models were also used to simulate both undulator ellipticities

=( )

*ue* 0, 0.34 for the same value of*a*¯*u*=5.0. The results are shown inﬁgure5. The simulations are seen to

agree well with each other in the linear regime with the elliptical undulator measured as having the shorter gain
length*lg*(0.34) *lg*( )0 »0.931, in good agreement with the value calculated from scaling.

A similar scaling argument for the electron pulse energies at saturation gives*E*3(0.34) *E*3( )0 =1.071which
is more difﬁcult to compare with the simulations ofﬁgure5due to the problem in deﬁning the points of
saturation. Note again, the difference in the simulation results between the averaged and unaveraged models as
saturation is approached and the fundamental interaction drives that of the harmonic. The divergence between
the two models is probably more pronounced in this case where*a*¯*u*=5.0, than that ofﬁgure4where*a*¯*u*=1.0.

### 4. Conclusion

An averaged FEL model in the 1D limit for elliptically polarised undulators including resonant radiation harmonics was presented. The undulator ellipticity changes the previous difference of Bessel functions factor, familiar from planar undulator FEL theory, into a more general elliptical Bessel function factor, valid for a planar undulator through to an helical undulator. This new elliptical factor was incorporated into a set of averaged, scaled, differential equations describing the FEL interaction. The scaling of these equations allows important quantities such as the gain length and radiation pulse energy, to be estimated as a function of the undulator ellipticity.

This averaged elliptical FEL model of the undulator was also solved numerically and the scaling
demonstrated. One notable result is that the harmonic gain and saturation energy for larger values of the
undulator parameter*a*¯*u*, was greater for elliptically polarised undulators than for the planar equivalent.

The averaged elliptical FEL model was also compared with the numerical simulations of an unaveraged FEL model using the Pufﬁn code which is also able to model elliptically polarised undulators(also in 3D). Overall, there was very good agreement between the two models. However, there were differences noted in the radiation pulse energy evolution of the harmonics as the interactions approached saturation and the harmonics are strongly coupled and driven by the interaction at the fundamental. This is not directly related to the ellipticity of the polarisation, but is thought to be a more general issue related to the validity of the averaging process in accurately describing the coupling between the fundamental and harmonic interactions. This topic will require further research.

Figure 5.Comparison of the scaled pulse energies for both averaged and unaveraged simulations of the third harmonicsE3in an undulator withaw=5.0 for two different undulator ellipticitiesue=0.0(planar undulator)andue=0.34(elliptical undulator). The

third harmonic interaction is seen to be stronger for the elliptical undulator, in agreement with the results ofﬁgure1, which shows that
the coupling parameter is maximum for the elliptical undulator case. The gain lengths of both results agree well with predicted scaling
via the elliptical coupling parameter*a*3.

8

### Acknowledgments

We gratefully acknowledge support of Science and Technology Facilities Council Agreement Number 4163192 Release#3; ARCHIE-WeSt HPC, EPSRC grant EP/K000586/1; EPSRC Grant EP/M011607/1; and John von Neumann Institute for Computing(NIC)on JUROPA at Jlich Supercomputing Centre(JSC), under project HHH20.

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